Optimization,Rajesh Kumar Arora 2015

• Optimization

• By: Rajesh Kumar Arora

• Publisher: CRC Press

• Pub. Date: May 19, 2015

• ISBN-13: 978-1-4987-2115-8

• Web ISBN-13: 978-1-4987-2115-8

• Print ISBN-13: 978-1-4987-2112-7

• Print ISBN-10: 1-4987-2112-5

• Pages in Print Edition: 466

• Subscriber Rating: [0 Ratings

  
  

1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code bisection.m
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % a-> lower bound of the design variable
   
6. % b> upper bound of the design variable
   
7. % alpha -> midpoint of a and b
   
8. % delx -> ?x for central difference method
   
9. % derivative -> derivative using central difference method
   
10. % derivative_alpha -> derivative at x = alpha
    
11. % abs -> absolute of a number, MATLAB function
    
12. %
    
13. clear all
    
14. clc
    
15. a = 40;
    
16. b = 90;
    
17. epsilon = 0.01;
    
18. delx = 0.01;
    
19. fprintf('    a             b     \n' )
    
20. fprintf('-------------------------\n' )
    
21. 
    
22. for i= 1:100
    
23. fprintf(' %7.3f      %8.3f \n',a,b)
    
24. alpha = (a+b)/2;
    
25. derivative = (func(a+delx) - func(a-delx) )/(2*delx);
    
26. derivative_alpha = (func(alpha+delx)- func(alpha-delx))/(2*delx);
    
27. if (derivative*derivative_alpha) < 0
    
28.                     b = alpha;
    
29. else
    
30.             a = alpha;
    
31. end
    
32. 
    
33. if abs(a-b) < epsilon
    
34.                     break;
    
35. end
    
36. end
    
37. 
    
38. fprintf('-------------------------\n' )
    
39. fprintf('x* =  %7.3f       Minimum =   %8.3f\n',a,func(a))
    
40. fprintf('Number of function calls =     %3d\n',4*i)
    
41. %
    
42. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code cubic
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % a-> lower bound of the design variable
   
6. % b> upper bound of the design variable
   
7. % alpha -> midpoint of a and b
   
8. % delx -> ?x for central difference method
   
9. % derivative -> derivative using central difference method
   
10. % derivative_alpha -> derivative at x = alpha
    
11. % abs -> absolute of a number, MATLAB function
    
12. % flag -> set the flag when minimum is bracketed
    
13. % derivative_a -> derivative at point a
    
14. % derivative_b -> derivative at b
    
15. %
    
16. a = 40;
    
17. b = 90;
    
18. delx = 0.01;
    
19. flag = 0;
    
20. epsilon= 0.001;
    
21. fprintf('    a             b     \n' )
    
22. fprintf('-------------------------\n' )
    
23. 
    
24. for i= 1:100
    
25. alpha = (a+b)/2;
    
26. derivative = (func(a+delx) - func(a-delx) )/(2*delx);
    
27. derivative_alpha = (func(alpha+delx)-func(alpha-delx) )/(2*delx);
    
28. if (derivative*derivative_alpha) < 0
    
29.     b = alpha;
    
30.     flag = 1;
    
31. else
    
32.     a = alpha;
    
33. end
    
34. if flag == 1
    
35.     break;
    
36. end
    
37. end
    
38. 
    
39. for j=1:100
    
40. fprintf(' %7.3f      %8.3f \n',a,b)
    
41. derivative_a = (func(a+delx) - func(a-delx) )/(2*delx);
    
42. derivative_b = (func(b+delx) - func(b-delx) )/(2*delx);
    
43. z = 3*(func(a)-func(b))/(b-a) + derivative_a + derivative_b;
    
44. w = ((b-a)/abs(b-a))*sqrt(z*z-derivative_a*derivative_b);
    
45. mew = (derivative_b+w-z)/(derivative_b-derivative_a+2*w);
    
46. if mew <= 1
    
47.     x_opt = b - mew*(b-a);
    
48. else
    
49.     x_opt = a;
    
50. end
    
51. alpha1 = (func(x_opt+delx) - func(x_opt-delx) )/(2*delx);
    
52. if abs(alpha1) < epsilon
    
53.     break;
    
54. else
    
55.    if (derivative_a*alpha1) < 0
    
56.     b = x_opt;
    
57. else
    
58.     a = x_opt;
    
59. end
    
60. end
    
61. 
    
62. end
    
63. 
    
64. fprintf('-------------------------\n' )
    
65. fprintf('x* =  %7.3f       Minimum =   %8.3f\n',x_opt,func(x_opt))
    
66. fprintf('Number of function calls =     %3d\n',4*i+8*j)
    
67. %
    
68. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code exhaustive.m
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % delta -> step size for search
   
6. % T -> independent variable, temperature
   
7. % U -> cost function
   
8. % uvec -> vector of cost function evaluated at
   
9. %         different temperatures
   
10. % minu -> minimum of cost function
    
11. % min -> MATLAB function
    
12. %
    
13. clear all
    
14. clc
    
15. uvec=[];
    
16. delta = 0.01;
    
17. for T = 40:delta:90
    
18.     U = 204165.5/(330-2*T) + 10400/(T-20);
    
19.     uvec = [uvec U];
    
20.     plot(T,U)
    
21.     hold on
    
22. end
    
23. xlabel('T');ylabel('U');
    
24. [minu,i]= min(uvec);
    
25. fprintf('Minimum Cost =  %6.2f\n ',minu)
    
26. fprintf('occurs at T =  %6.2f\n ',40+(i-1)*delta)
    
27. 
    
28. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code golden.m
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % a-> lower bound of the design variable
   
6. % b> upper bound of the design variable
   
7. % alpha -> midpoint of a and b
   
8. % falpha1 -> function value at x = alpha1
   
9. % falpha2 -> function value at x = alpha2
   
10. % epsilon -> constant used to terminate the algorithm
    
11. % abs -> absolute of a number, MATLAB function
    
12. % tau -> 2-golden number
    
13. %
    
14. clear all
    
15. clc
    
16. a = 40;
    
17. b = 90;
    
18. epsilon = 0.00001;
    
19. tau = 0.381967;
    
20. alpha1 = a*(1-tau) + b*tau;
    
21. alpha2 = a*tau + b*(1-tau);
    
22. falpha1 = func(alpha1);
    
23. falpha2 = func(alpha2);
    
24. 
    
25. fprintf('    a             b     \n' )
    
26. fprintf('-------------------------\n' )
    
27. 
    
28. for i= 1:100
    
29.     fprintf(' %7.3f      %8.3f \n',a,b)
    
30.     if falpha1 > falpha2
    
31.         a = alpha1;
    
32.         alpha1 = alpha2;
    
33.         falpha1 = falpha2;
    
34.         alpha2 = tau*a + (1-tau)*b;
    
35.         falpha2 = func(alpha2);
    
36.     else
    
37.         b = alpha2;
    
38.         alpha2 = alpha1;
    
39.         falpha2 = falpha1;
    
40.         alpha1 = tau*b + (1-tau)*a;
    
41.         falpha1 = func(alpha1);
    
42.     end
    
43. 
    
44. if abs(func(alpha1)-func(alpha2))< epsilon
    
45.     break;
    
46. end
    
47. end
    
48. 
    
49. fprintf('-------------------------\n' )
    
50. fprintf('x* =  %7.3f       Minimum =   %8.3f\n',alpha1,func(alpha1))
    
51. fprintf('Number of function calls =     %3d\n',2+i)
    
52. %
    
53. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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1. A%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code newtonraphson.m
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % x -> initial guess of design variable
   
6. % delx -> ?x for central difference method
   
7. % derivative -> derivative using central difference method
   
8. % sec_derivative -> second derivative
   
9. % epsilon -> constant used to terminate the program
   
10. % xprev -> value of x stored from previous iteration
    
11. %
    
12. clear all
    
13. clc
    
14. x = 45;
    
15. delx = 0.01;
    
16. epsilon = 0.01;
    
17. fprintf('     x    f(x)      Deriv. Second deriv.\n' )
    
18. fprintf('-----------------------------------------\n' )
    
19. 
    
20. for i= 1:100
    
21. derivative = (func(x+delx) - func(x-delx) )/(2*delx);
    
22. sec_derivative=(func(x+delx)+func(x-delx)-2*func(x))/(delx*delx);
    
23. fprintf('%8.3f %8.3f %8.3f %8.3f\n',x,func(x),derivative,      sec_derivative)
    
24. xprev = x;
    
25. x = x- derivative/sec_derivative;
    
26. if abs(x-xprev) < epsilon
    
27.     break;
    
28. end
    
29. end
    
30. 
    
31. fprintf('-----------------------------------------\n' )
    
32. fprintf('Number of function calls =     %3d\n',5*i)
    
33. 
    
34. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code grad_vec.m
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. function deriv = grad_vec_complex(x,delx,n_of_var)
   
6. xvec = x;
   
7. h = 1e-14;
   
8. for j = 1:length(x)
   
9. xvec = x;
   
10. c = complex(xvec(j),h);
    
11. xvec(j) = c;
    
12. deriv(j) = imag(func_multivar(xvec)/h);
    
13. end
    
14. %
    
15. %%%%%%%%%%%%%%%%%

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1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code levenberg-marquardt.m
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % n_of_var -> number of design variables
   
6. % x = [-1.5 1.5] -> starting value of x
   
7. % lambda -> initially set to a large value
   
8. % epsilon1, epsilon2 -> constant used for terminating
   
9. %                       the algorithm
   
10. % delx -> required for gradient computation
    
11. % f_prev -> function valye at first / previous iteration
    
12. % deriv -> gradient vector
    
13. % sec_deriv -> hessian matrix
    
14. % search -> search direction (vector)
    
15. %
    
16. clear all
    
17. clc
    
18. n_of_var = 2;
    
19. x = [-3 2];
    
20. lambda = 1e3;
    
21. epsilon1 = 1e-7;
    
22. epsilon2 = 1e-7;
    
23. delx = 1e-3;
    
24. f_prev = func_multivar(x);
    
25. fprintf('Initial function value =  %7.4f\n ',f_prev)
    
26. fprintf(' No.       x-vector      f(x)      Deriv \n')
    
27. fprintf('__________________________________________\n')
    
28. for i = 1:100
    
29.     f_prev = func_multivar(x);
    
30.     deriv = grad_vec(x,delx,n_of_var);
    
31.     sec_deriv = hessian(x,delx,n_of_var);
    
32.     search = -inv(sec_deriv+lambda*eye(length(x)))*deriv';
    
33.     x = x + search';
    
34.     f = func_multivar(x);
    
35.         if f < f_prev
    
36.         lambda = lambda/2;
    
37.             else
    
38.         lambda = 2*lambda;
    
39.             end
    
40. 
    
41.             if abs(f-f_prev)<epsilon1 || norm(deriv)<epsilon2
    
42.         break;
    
43.        end
    
44. fprintf('%3d %8.3f %8.3f  % 8.3f  %8.3f  \n',i,x,f,norm(deriv))
    
45. end
    
46. 
    
47. fprintf('__________________________________________\n')
    
48. %
    
49. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
2. % MATLAB code levenberg-marquardt.m
   
3. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
4. %
   
5. % n_of_var -> number of design variables
   
6. % x = [-1.5 1.5] -> starting value of x
   
7. % lambda -> initially set to a large value
   
8. % epsilon1, epsilon2 -> constant used for terminating
   
9. %                       the algorithm
   
10. % delx -> required for gradient computation
    
11. % f_prev -> function valye at first / previous iteration
    
12. % deriv -> gradient vector
    
13. % sec_deriv -> hessian matrix
    
14. % search -> search direction (vector)
    
15. %
    
16. clear all
    
17. clc
    
18. n_of_var = 2;
    
19. x = [-3 2];
    
20. lambda = 1e3;
    
21. epsilon1 = 1e-7;
    
22. epsilon2 = 1e-7;
    
23. delx = 1e-3;
    
24. f_prev = func_multivar(x);
    
25. fprintf('Initial function value =  %7.4f\n ',f_prev)
    
26. fprintf(' No.       x-vector      f(x)      Deriv \n')
    
27. fprintf('__________________________________________\n')
    
28. for i = 1:100
    
29.     f_prev = func_multivar(x);
    
30.     deriv = grad_vec(x,delx,n_of_var);
    
31.     sec_deriv = hessian(x,delx,n_of_var);
    
32.     search = -inv(sec_deriv+lambda*eye(length(x)))*deriv';
    
33.     x = x + search';
    
34.     f = func_multivar(x);
    
35.         if f < f_prev
    
36.         lambda = lambda/2;
    
37.             else
    
38.         lambda = 2*lambda;
    
39.             end
    
40. 
    
41.             if abs(f-f_prev)<epsilon1 || norm(deriv)<epsilon2
    
42.         break;
    
43.        end
    
44. fprintf('%3d %8.3f %8.3f  % 8.3f  %8.3f  \n',i,x,f,norm(deriv))
    
45. end
    
46. 
    
47. fprintf('__________________________________________\n')
    
48. %
    
49. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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